Comparison between ISRA and RLA algorithms.: Use of a Wiener Filter based stopping criterion

This paper consists of two parts. In the first one, we make a comparative analysis of the algorithms of Richardson-Lucy (RLA) and the Image Space Reconstruction Algorithm (ISRA), for image deconvolution of astronomical images. These iterative algorithms keep the reconstructed image non-negative while maximizing the likelihood for a Poisson Process (RLA) or a Gaussian process (ISRA). Their comparison is made easier when these algorithms are rewritten as descent algorithms; the additive forms evidence the role of the variances of the noise and allow a better understanding of these algorithms. A numerical illustration is performed from simulated images. In practice, the results obtained by the two algorithms appeared to be very similar, independently of the statistics of the noise. In the second part of the paper, we propose a new objective-stopping technique that makes use of a comparison of the results of these algorithms with that of the Wiener filter. The comparison is made in the Fourier plane, computing the Euclidean distances between modules of the spatial frequencies components of the images. We propose then to use the results of ISRA and RLA at the iteration number corresponding to the minimum of that distance. The technique is checked in a numerical simulation, for which the optimal iteration numbers can be easily determined. A good agreement is obtained between best ISRA and RLA results and the Wiener filter, in particular for rather noisy images. However ISRA and RLA produce very high frequency components, outside the cut-off frequency of the instrument; limiting the iteration number alone cannot allow a perfect agreement with the Wiener approach, stressing the need of an explicit regularization for astronomical applications.